Multivariate Calculus - Bruce E. Shapiro

LECTURE 17. DOUBLE INTEGRALS OVER GENERAL REGIONS 143
Figure 17.8: The solid formed by the paraboloid z = 9 − x 2 − y 2 and the xy plane.
xy plane. This occurs when z = 0. So that means the domain is
0 = 9 − x 2 − y 2 ⇒ y 2 + x 2 = 3 2
which is a circle of radius 3 centered at the origin. Since we only want the part of
the circle that is in the first quadrant, we need x and y both positive. To cover the
upper right hand corner of a circle of radius 3 centered at the origin, we can let x
increase from 0 to 3 and y increase from 0 to y = √ 9 − x 2 Therefore the volume is
V =
=
=
=
= 2 3
∫ 3 ∫ √ 9−x 2
0
∫ 3
0
∫ 3
0
∫ 3
0
(9 − x 2 )
(9 − x 2 − y 2 )dydx
∫ √ 9−x 2
0
(
(9 − x 2 ) y|
0
∫ 3
√
9−x 2
0
(9 − x 2 ) 3/2 dx − 1 3
0
(9 − x 2 ) 3/2 dx
dydx −
)
dx − 1 3
∫ 3
0
∫ 3 ∫ √ 9−x 2
0 0
∫ 3
0
(9 − x 2 ) 3/2 dx
√
9−x 2
y 2 dydx
y 3 | 0 dx
Math 250, Fall 2006 Revised December 6, 2006.